Research Article Fatigue Failure Model for Polymeric ...
Transcript of Research Article Fatigue Failure Model for Polymeric ...
Hindawi Publishing CorporationISRN Polymer ScienceVolume 2013, Article ID 321489, 11 pageshttp://dx.doi.org/10.1155/2013/321489
Research ArticleFatigue Failure Model for Polymeric Compliant Systems
Theddeus T. Akano and Omotayo A. Fakinlede
Department of Systems Engineering, University of Lagos, Akoka, Lagos 101017, Nigeria
Correspondence should be addressed toTheddeus T. Akano; [email protected]
Received 21 January 2013; Accepted 28 February 2013
Academic Editors: H. M. da Costa, A. Mousa, and A. Uygun
Copyright Β© 2013 T. T. Akano and O. A. Fakinlede. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
Fatigue analysis and lifetime evaluation are very important in the design of compliant mechanisms to ensure their safety andreliability. Earlier models for the fatigue prediction of compliant mechanisms are centred on repeated and reversed stress cycles.Compliant mechanisms (CMs) are now being applied to situations where the fatigue is caused by random varying stress cycles. It is,therefore, necessary to consider fatigue resulting from random varying stress cycles and damage caused to the compliant material.A continuum damage mechanics (CDM) model is proposed to assess the fatigue life of polymeric compliant mechanisms. Theelastic strain energy is computed on the basis of a nearly incompressive hyperelastic constitution. The damage evolution equationis used to develop a mathematical formula that describes the fatigue life as a function of the nominal strain amplitude under cyclicloading. Low density polypropylene (LDP) is used for the fatigue tests conducted under displacement controlled condition with asine waveform of 10Hz.The results from the theoretical formula are comparedwith those from the experiment and fatigue software.The result from the prediction formula shows a strong agreement with the experimental and simulation results.
1. Introduction
Fatigue is one of themajor failuremechanisms in engineeringstructures [1]. Time-varying cyclic loads result in failureof components at stress values below the yield or ultimatestrength of the material. Fatigue failure of components takesplace by the initiation and propagation of a crack until itbecomes unstable and then propagates to sudden failure.The total fatigue life is the sum of crack initiation life andcrack propagation life. Fatigue life prediction has becomeimportant because of the complex nature of fatigue as itis influenced by several factors, statistical nature of fatiguephenomena and time-consuming fatigue tests.
Though a lot of fatigue models have been developed andused to solve fatigue problems, the range of validity of thesemodels is not well defined. No method would predict thefatigue life with the damage value by separating crack initi-ation and propagation phases. The methods used to predictcrack initiation life are mainly empirical [2] and they fail todefine the damage caused to the material. Stress- or strain-based approaches followed do not specify the damage causedto the material, as they are mainly curve fitting methods.The limitation of this approachmotivated the development of
micromechanicsmodels termed as local approaches based oncontinuum damage mechanics (CDM).The local approachesare based on application of micromechanics models offracture in which stress/strain and damage at the crack tip arerelated to the critical conditions required for fracture. Thesemodels are calibrated through material specific parameters.Once these parameters are derived for particular material,they can be assumed to be independent of geometry andloading mode and may be used to the assessment of acomponent fabricated from the same material.
For some compliant structures, the desired motion mayoccur infrequently, and the static theories may be enoughfor the analysis [3]. However, by the definition of compliantmechanisms, deflection of flexible members is required forthe motion. Usually, it is desired that the mechanism becapable of undergoing the motion many times, and designrequirements may be many millions of cycle of infinitelife. This repeated loading cause fluctuating stresses in themembers and can result in fatigue failure. Failure can occurat stresses that are significantly lower than those that causestatic failure [3]. A small crack is enough to initiate thefatigue failure. The crack progresses rapidly since the stressconcentration effect becomes greater around it. If the stressed
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area decreases in size, the stress increases in magnitude, andif the remaining area is small, themember can fail. Amemberfailed because of fatigue showing two distinct regions. Thefirst one is due to the progressive development of the crack,while the other one is due to the sudden fracture. Prematureor unexpected failure of a device can result in an unsafedesign.The consumer confidencemay be reduced in productsthat fail prematurely. For these and other reasons, it is criticalthat the fatigue life of compliant mechanism be analyzed.
Although fatigue failure is difficult to predict accurately,an understanding of fatigue failure prediction and preventionis very helpful in the design of compliant mechanisms. Thetheory can be used to design devices that will withstand thesefluctuating stresses.
Several models are available for fatigue failure prediction.The stress-life and strain-life models are commonly used inthe design of mechanical components [3]. These theories areappropriate for parts that undergo consistent and predictablefluctuating stresses. Many machine components fit into thiscategory because their motion and loads are defined bykinematics of the mechanism. There are three stress cycleswith which loads may be applied to the component underconsideration. The simplest being the reversed stress cycle(Figure 1(a)). This is merely a sine wave where the maximumstress and minimum stress differ by a negative sign. Anexample of this type of stress cycle would be in an axle,in which every half turn or half period as in the case ofthe sine wave, the stress on a point would be reversed. Themost common type of cycle found in engineering applicationsis where the maximum stress and minimum stress areasymmetric (the curve is a sinewave), not equal, and opposite(Figure 1(b)). This type of stress cycle is called repeated stresscycle. A final type of cyclemode is where stress and frequencyvary randomly (Figure 1(c)). An example of this would behull shocks, where the frequencymagnitude of the waves willproduce varying minimum and maximum stresses.
Predicting the life of parts stressed above the endurancelimit is at best a rough procedure. For the large percentageof mechanical and structural compliant systems subjectedto randomly varying stress cycle intensity (e.g., compliantautomotive suspension and compliant aircraft structuralcomponents, etc.), the prediction of fatigue life is furthercomplicated. The normal stress-life and strain-life modelscannot be adopted in the fatigue prediction. Models such ascontinuumdamagemechanics (CDM) can be used in dealingwith this situation.
Polymers are predominantly used in the design of com-pliant mechanisms [3]. It is important to use the nonlinearcharacteristics of polymers to analyse the performance ofcompliant systems. Thermoplastic polymers like polypropy-lene exhibit a viscoelastic material response [4]. It has beenfrequently noted that with certain constitutive laws, such asthose of viscoelasticity and associative plasticity, the materialbehaves in a nearly incompressible manner [5]. The typicalvolumetric behavior of hyperelastic materials can be groupedinto two classes. Materials such as polymers typically havesmall volumetric changes during deformation and those thatare incompressible or nearly-incompressible materials [6].An example of the second class of materials is foams, which
can experience large volumetric changes during deformation,and these are compressible materials. This implies that mostpolymers are nearly incompressible. In general, the responseof a typical polymer is strongly dependent on temperature[7]. At low temperatures, polymers deform elastically likeglass; at high temperatures, the behaviour is viscous likeliquids; at moderate temperatures, the behaviour is like arubbery solid. Hyperelastic constitutive laws are intended toapproximate this rubbery behaviour. Polymers are capable oflarge deformations and subject to tensile and compressionstress-strain curves [8]. The simplest yet relatively precisedescription for this type of material is isotropic hyperelastic-ity [8].
The fatigue failure of thermoplastics polymers generallydevelops in two phases [9]. First, the material accumulatesfatigue damage (i.e., in the initiation phase), which ultimatelyleads to the formation of visible crazes. The crazes furthergrow, form cracks and propagate (i.e., in the propagationphase) until the final failure occurs. In general, the damageprocess in polymers is regarded as the formation and devel-opment of microdefects and crazes within an initially perfectmaterial. The material remains the same but its macroscopicproperties change with its microscopic geometry [10]. Inpolymers, craze formation is generally believed to be one ofthe main causes of material damage, which is both a localizedyielding process and the first stage of fracture. Crazes areusually initiated either at surface flaws and scratches orat internal voids and inclusions and affect significantly thesubsequent deformation and bulk mechanical behaviors ofpolymers [11]. The continuum damage mechanics (CDM)first introduced by Kachanov and developed within theframework of thermodynamics discusses systematically theeffects of microdefects on the subsequent development ofmicrodefects and the states of stress and strain inmaterials. Ithas been applied to fatigue and fracture of different materials.
In this paper, an isotropic damage evolution equationfor finite viscoelasticity characteristic of polymeric CMs isproposed, which is based on the CDM. A new damage modelis developed to establish the fatigue life formula for suchcompliant systems. The compliant material is idealized as anisotropic hyperelastic material. A commonly used polymericmaterial, low density polypropylene (LDP) was tested toobtain the fatigue life as a function of the strain amplitude.
2. The Literature Review
A few researchers have looked into the fatigue failure ofcompliant systems. Li et al. [12] used the modified Basquinequation to determine the life cycle till failure for compliantfast tool servo. The fatigue life according to the equationis a function of the equivalent reverse stress, fatigue stressconcentration factor, the range stress, the ultimate strength,the average stress, and the endurance limit. Demirel et al.[13] and Subasi [14] used the factor of safety expressedin terms of the fluctuating stresses, endurance limit, meanstress component, and an alternating stress component forfatigue failure prediction of compliant mechanism. If thestress condition is below the two lines described in modified
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Stre
ss
Cycles
πmax
πmin
ππ
Tens
ion+
βC
ompr
essio
n
(a) Reversed
Cycles
πmin
ππ
0Stre
ssTe
nsio
n+β
Com
pres
sion
(b) Repeated
Cycles
Stre
ssTe
nsio
n+β
Com
pres
sion
(c) Random
Figure 1: Stress cycles showing (a) reversed, (b) repeated, and (c) random cycles.
Goodman diagram for fatigue failure, the compliant memberis expected to have an infinite life. Howell et al. [15] proposeda method for the probabilistic design of a bistable compliantslider-crank mechanism which its objective function is themaximization of themechanism reliability in fatigue. Cannonet al. [16] used the modified fatigue strength at cycles whichis expressed in terms of Marin correction factors and thetheoretical fatigue strength to predict the failure behaviourof a compliant end-effector for microscribing. This gives theS-N diagram for the mechanism where the maximum stressis compared with the modified fatigue strength.
Quite a number of researchers have employed the conceptof damage evolution in the prediction of fatigue failure ofengineering structures and components. Jiang [17] deriveda damaged evolution model for strain fatigue of ductilemetals based on Lemaitreβs potential of dissipation. Thenthe equation of fatigue life prediction and the criterion ofcumulative fatigue damage were deduced. The model wasvalidated with experiment. Shi et al. [18] proposed a newdamage mechanics model to predict the fatigue life of fiberreinforced polymer lamina and adopted the singularity ofstiffnessmatrix as the failure criterion of lamina in this article,which inventively transformed the complex anisotropic issueof composite lamina fatigue into the analysis of single-variable isotropic damages for fiber and matrix. Akshantalaand Talreja [19] proposed a methodology for fatigue-lifeprediction that utilizes a micromechanics-based evaluationof damage evolution in conjunction with a semiempiricalfatigue failure criterion. The specific case treated was that of
crossply laminates under cyclic tension.The predicted resultswere comparedwith experimental data for several glass epoxyand carbon epoxy laminates. Ping et al. [20] proposed anonlinear continuum damage mechanics model to assessthe creep-fatigue life of a steam turbine rotor, in whichthe effects of complex multiaxial stress and the coupling offatigue and creep are taken into account. The results werecompared with those from the linear accumulation theorythat had been dominant in life assessment of steam turbinerotors. The comparison shows that the nonlinear continuumdamage mechanics model describes the accumulation anddevelopment of damage better than the linear accumulationtheory. Ali et al. [21] investigated the fatigue behaviourof rubber using dumb-bell test specimens under uniaxialloading. Inmodeling fatigue damage behaviour, a continuumdamage model was presented based on the function of thestrain range under cyclic loading. Upadhyaya and Sridhara[22] predicted strain controlled fatigue life of EN 19 steel and6082-T6 aluminum alloy considering both crack initiationand crack propagation phases. The theory of continuumdamage mechanics was used in the study of fatigue damagephenomena such as the nucleation and initial defect growth(microvoids and microcracks) in elastomers by Wang et al.[23] and Mahmoud et al. [24].
Continuum damage mechanics (CDM) approach modelsthe crack initiation life with a damage value, and damagebeyond the crack initiation phase is predicted by fracturemechanics in terms of crack size. Fatigue life was predictedbased on this concept.
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3. Fatigue Failure Prediction Model
Finite element implementations of nearly incompressiblematerial models often employ decoupled numerical treat-ments of the dilatation and deviatoric parts of the deforma-tion gradient. The strain energy density function for suchmaterial is decoupled as
Ξ¨ (πΆ) = Ξ¨ (πΆ) + U (π½) , (1)
where
Ξ¨(C) = ππ (tr C β 3) ,
U (π½) = π½(π½ β 1)2,
ππ =1
2
π
β
π=1
ππ; π½ =ππ
2,
(2)
π and π are the material properties known as bulk and shearmodulus, respectively; π½ is the Jacobian determinant of thedeformation gradient. The strain energy density could beexpressed in terms of the principal stretches ππ as
Ξ¨ (π1, π2, π3) =
π
β
π=1
ππ
πΌπ
(ππΌπ
1+ ππΌπ
2+ ππΌπ
3β 3) +
π
β
π=1
π½(π½ β 1)2.
(3)
The principal components of the Cauchy stress are given by[25]
ππ =ππ
π1π2π3
πΞ¨
ππ1
, π = 1, 2, 3. (4)
The strain energy potential can be written as either functionof the principal stretch ratios or as a function of the invariantsof the strain tensor πΆ, πΌ1, πΌ2, πΌ3. The invariants of C aredefined as
πΌ1 = trC = C : I,
πΌ2 = trCC,
πΌ3 = detC = π½2.
(5)
In terms of the principal stretch ratio, the invariants arewritten as
πΌ1 = π2
1+ π2
2+ π2
3, (6)
πΌ2 = π2
1π2
2+ π2
2π2
3+ π2
3π2
1, (7)
πΌ2 = π2
1π2
2π2
3. (8)
The invariants could be expressed in terms of the deviatoricprincipal stretches as
πΌ1 = π½β2/3
πΌ1,
πΌ2 = π½β4/3
πΌ2,
πΌ3 = π½2.
(9)
Substituting (6) and (8) into (1) gives
Ξ¨ =
π
β
π=1
ππ
πΌπ
(π½β2/3
(ππΌπ
1+ ππΌπ
2+ ππΌπ
3) β 3) +
π
β
π=1
π½(π½ β 1)2.
(10)
Substituting (10) into (4) gives
ππ
ππ½
πππ
=
π
β
π=1
ππ
πΌπ
(β2
3π½β5/3
ππ
ππ½
πππ
(ππΌπ
1+ ππΌπ
2+ ππΌπ
3)
+π½β2/3
πΌπππΌπ
π)
+
π
β
π=1
2π½ (π½ β 1) ππ
ππ½
πππ
.
(11)
Since π½ = π2
1π2
2π2
3, we have
ππ
ππ½
πππ
= π2
1π2
2π2
3= π½. (12)
Therefore,
ππ
πΞ¨
πππ
= [
π
β
π=1
ππ
πΌπ
π½β2/3
(πΌπππΌπ
πβ
2
3(ππΌπ
1+ ππΌπ
2+ ππΌπ
3))
+
π
β
π=1
2π½π½ (π½ β 1)] .
(13)
The principal Cauchy stresses are, therefore, given as
ππ = [
π
β
π=1
ππ
πΌπ
π½β5/3
(πΌπππΌπ
πβ
2
3(ππΌπ
1+ ππΌπ
2+ ππΌπ
3))
+
π
β
π=1
2π½π½ (π½ β 1)] .
(14)
Then, the difference between the principal stresses becomes
π1 β π3 =
π
β
π=1
πππ½β5/3
(ππΌπ
1β ππΌπ
3) ,
π2 β π3 =
π
β
π=1
πππ½β5/3
(ππΌπ
2β ππΌπ
3) ,
π1 β π2 =
π
β
π=1
πππ½β5/3
(ππΌπ
1β ππΌπ
2) .
(15)
Since force is applied in one direction in most compliantmechanisms, we will consider a mechanism undergoinguniaxial stress state. The principal stretches become
π1 = π; π2 = π3 =β
π½
π, (16)
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π is the stretch in the loading direction; π2 and π3 arethe principal stretches on plane perpendicular to loadingdirection. Substituting (16) into (15) gives
π1 β π3 = π1 β π2 =
π
β
π=1
πππ½β5/3
(ππΌπ β (
π½
π)
πΌπ/2
) ,
π2 β π3 = 0.
(17)
The effective stress is given by the equation tensor
ππ = (1
2(π1 β π2)
2+ (π2 β π3)
2+ (π1 β π3)
2)
1/2
= (3
2ππππππ)
1/2
,
(18)
where πππ are the components of the deviator tensor πdev
πdev = π β1
3(tr π) πΌ. (19)
Substituting (17) into (19) gives
ππ =
π
β
π=1
πππ½β5/3
(ππΌπ β (
π½
π)
πΌπ/2
) . (20)
The stretch ratio in the loading direction is given by [26]
π = 1 + π, (21)
π is the nominal strain. Substituting (21) into (20) gives
ππ =
π
β
π=1
πππ½β5/3
((1 + π)πΌπ β (
π½
(1 + π))
πΌπ/2
) . (22)
Substituting (16) into (10), we have
Ξ¨ =
π
β
π=1
ππ
πΌπ
(π½β2/3
(ππΌπ + 2(
π½
π)
πΌπ/2
) β 3) +
π
β
π=1
π½ (π½ β 1)
=
π
β
π=1
ππ
πΌπ
(π½β2/3
((1 + π)πΌπ + 2(
π½
(1 + π))
πΌπ/2
) β 3)
+
π
β
π=1
π½ (π½ β 1) .
(23)
3.1. Continuum Damage Mechanics Model. Material damageusually induces the stiffness change of thematerial.Therefore,the damage state can be characterized by the change ofelastic constants. Consider a representative volume elementof an anisotropic material with stiffness [πΈ] damaged undera system of loading {ππ}. The stiffness matrix of the damagedmaterial is [πΈπ]. Then, the damage matrix [π·] is defined as[27]
[π·] = [πΌ] β [πΈπ] [πΈ]β1
, (24)
where [πΌ] is the identity matrix. The strain {π} is given as
{π} = [πΌ] β [πΈπ]β1
{ππ}
= [πΈ]β1
[[πΌ] β [π·]]β1
{ππ}
= [πΈ]β1
{ππ} ,
(25)
where
{ππ} = [[πΌ] β [π·]]β1
{ππ} . (26)
The matrix {ππ} is defined as the effective stress matrix aftermaterial damage. Hence, the damage effect matrix [π] is
[π] = [[πΌ] β [π·]]β1
. (27)
3.1.1. Isotropic Model. Consider a damaged body as shownin Figure 2, in which a representative volume element(RVE) is isolated. Damaged variable is physically definedby the surface density of microcracks and intersections ofmicrovoids lying on a plane cutting RVE of cross-section πΏπ΄
[28]. Damaged variableπ·(π), for the plane defined by normalπ, is
π·(π) =πΏπ΄π·
πΏπ΄, 0 β€ π·(π) β€ 1, (28)
where πΏπ΄π· is the effective area of the intersection of allmicrocavities or microcracks that lie in the initial area πΏπ΄ attime π‘. An isotropic damage variable is equally distributed inall directions, which is defined as
π· =πΏπ΄π·
πΏπ΄. (29)
Equation (29) is the percentage of the damaged area to initialarea.π· is a scalar. Isotropic damage is assumed in this conceptof continuumdamagemechanics.The damage parameter canbe obtained by reducing the rank of matrix to zero. Thisreduces (24) to
π· = 1 β πΈπ β πΈβ1
. (30)
The effective stress after material damage, ππ, and damagestrain energy release rate are related by [29]
ππ = π β ππ =ππ
1 β π·. (31)
The nominal stress-strain relation of a damage material is thesame in form as that of an undamaged material [30]. Thismeans that
ππ = ππ, (32)
so that
ππ
1 β π·=
π
β
π=1
πππ½β5/3
(ππΌπ β π½
πΌπ/2
πβπΌπ/2
) . (33)
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π(π‘)
πΏπ΄
πΏπ΄π·
π΄
π
RVE
(a) (b)
Figure 2: Transformation π(π‘) from the (a) initial undamaged configuration to (b) the damaged configuration.
The constitutive equation for damage evolution is given by[28]
οΏ½οΏ½ = βππΌ0
πY, (34)
πΌ0 is the dissipation potential;Y is the damage strain energyrelease rate; οΏ½οΏ½ is the damage growth rate. The dissipationpotential is assumed as [17]
πΌ0 =π 0
π0 + 1(βY
π 0
)
π0+1
, (35)
π0 and π 0 are material parameters determined by the exper-imental fatigue life as a function of the strain range. But thestrain energy of a damaged and undamaged material is thesame [27]. From the CDM theory, the damaged strain energyshould be a function of the effective nominal normal stressof the damaged configuration. Hence, in the uniaxial stressstate, the damaged strain energy released rateY is defined as
βY =πΞ¨
ππ·=
πΞ¨ (ππ)
ππ·. (36)
Substituting (23) into (36), we have
βY =πΞ¨ (ππ)
ππ·=
πΞ¨
ππ
ππ (ππ)
ππ·
=
π
β
π=1
ππ
πΌπ
π½β2/3
(πΌπππΌπβ1β πΌππ½
πΌπ/2πβ(πΌπ/2+1))
ππ (ππ)
ππ·.
(37)
Taking the partial derivative of (33) with respect to π·, gives
ππ (ππ)
ππ·
=
π
β
π=1
1
ππ
ππ
(1 β π·)2[π½β5/3
(πΌπππΌπβ1 +
πΌπ
2π½πΌπ/2πβ(πΌπ/2+1))]
β1
.
(38)
Substituting (38) into (37), we have
βY =
π
β
π=1
1
πΌπ
ππ
(1 β π·)2π½
Γ [ (πΌπππΌπβ1 β πΌππ½
πΌπ/2πβ(πΌπ/2+1))
Γ (πΌπππΌπβ1 +
πΌπ
2π½πΌπ/2πβ(πΌπ/2+1))
β1
] .
(39)
Substituting (33) into (39), we have
βY =
π
β
π=1
ππ
πΌπ
[
[
(ππΌπ β π½
πΌπ/2
πβπΌπ/2
)
(1 β π·)π½β2/3
Γ [ (πΌπππΌπβ1 β πΌππ½
πΌπ/2πβ(πΌπ/2+1))
Γ (πΌπππΌπβ1 +
πΌπ
2π½πΌπ/2πβ(πΌπ/2+1))
β1
]]
]
.
(40)
Using (34), (35), and (40), we have
οΏ½οΏ½ =
π
β
π=1
ππ
πΌπ
(π β1
0)[
[
(ππΌπ β π½
πΌπ/2
πβπΌπ/2
)
(1 β π·)π½β2/3
Γ [ (πΌπππΌπβ1 β πΌππ½
πΌπ/2πβ(πΌπ/2+1))
Γ (πΌπππΌπβ1 +
πΌπ
2π½πΌπ/2πβ(πΌπ/2+1))
β1
]]
]
π0
.
(41)
Under a cyclic loading condition, the damagewill accumulatewith the number of cycles, and the damage evolution willdepend on the strain amplitude. The time rate change of
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damage variable οΏ½οΏ½ can be represented in terms of theevolution of π· with respect to the number of cycles. Basedon this consideration, the principal stretch amplitude Ξπ isused to replace π, the fatigue damage evolution per cycle isthen expressed asππ·
ππ
=
π
β
π=1
ππ
πΌπ
(π β1
0)[
[
(ΞππΌπ β π½
πΌπ/2
ΞπβπΌπ/2
)
(1 β π·)π½β2/3
Γ [(πΌπΞππΌπβ1 β πΌππ½
πΌπ/2Ξπβ(πΌπ/2+1))
Γ (πΌπΞππΌπβ1+
πΌπ
2π½πΌπ/2Ξπ
β(πΌπ/2+1))
β1
]]
]
π0
,
(42)
where Ξπ is the principal stretch amplitude and π is thenumber of cycles. Assuming that the damage variable π· iszero at the beginning of the cyclic loading, that is,π· = 0whenπ = 0, then the damage value at any cycle can be determinedby integrating (42), which gives
β«
π·
0
(1 β π·)π0ππ·
= β«
π
0
[
π
β
π=1
ππ
πΌπ
(π β1
0) π½β2/3
Γ [ (ΞππΌπ β π½
πΌπ/2
ΞπβπΌπ/2
)
Γ (πΌπΞππΌπβ1 β πΌππ½
πΌπ/2Ξπβ(πΌπ/2+1))
Γ (πΌπΞππΌπβ1 +
πΌπ
2π½πΌπ/2Ξπ
β(πΌπ/2+1))
β1
]
π0
]ππ.
(43)
The relation between the damage variable π· and number ofcycles π could be deduced as
[1 β (1 β π·)π0+1]
= (π0 + 1)
Γ [
π
β
π=1
ππ
πΌπ
(π β1
0) π½β2/3
Γ [ (ΞππΌπ β π½
πΌπ/2
ΞπβπΌπ/2
)
Γ (πΌπΞππΌπβ1 β πΌππ½
πΌπ/2Ξπβ(πΌπ/2+1))
Γ (πΌπΞππΌπβ1 +
πΌπ
2π½πΌπ/2Ξπ
β(πΌπ/2+1))
β1
]
π0
]π.
(44)
The damage variable is expressed as the ratio of the numberof cycles π to the fatigue life ππ as [17]
π· =π
ππ
. (45)
Equation (45) indicates that the damage is linearly distributedto each cycle during the loading. Therefore, if a material hasbeen subjected to cyclic loading, the damage is
π· =ππ
(ππ)π
, (46)
and when the fatigue rupture occurs, we have
π· = π·π = 1, (47)
π·π is the critical value of the damage variable. Equations (45)to (47) show that at the moment of failure
π = ππ, (48)
and the fatigue life ππ is expressed as
ππ = (π0 + 1)β1
Γ [
π
β
π=1
ππ
πΌπ
(π β1
0)
Γ [ (ΞππΌπ β π½
πΌπ/2
ΞπβπΌπ/2
)
Γ (πΌπΞππΌπβ1 β πΌππ½
πΌπ/2Ξπβ(πΌπ/2+1))
Γ (πΌπΞππΌπβ1 +
πΌπ
2π½πΌπ/2Ξπ
β(πΌπ/2+1))
β1
]]
βπ0
.
(49)
Using (21), the principal stretch amplitude, Ξπ, and thenominal strain amplitude, Ξπ, are related as
Ξπ = 1 + Ξπ. (50)
Substituting (50) into (49), the formula for the fatigue life isexpressed as a function of the nominal strain amplitude fora compliant mechanism under large deformation and cyclicloading
ππ
= (π0 + 1)β1
[
π
β
π=1
ππ
πΌπ
(π β1
0)
Γ [ ((1 + Ξπ)πΌπ β π½
πΌπ/2(1 + Ξπ)
βπΌπ/2)
Γ (πΌπ(1 + Ξπ)πΌπβ1
βπΌππ½πΌπ/2
(1 + Ξπ)β(πΌπ/2+1))
8 ISRN Polymer Science
Γ (πΌπ(1 + Ξπ)πΌπβ1
+
πΌπ
2π½πΌπ/2
(1 + Ξπ)β(πΌπ/2+1))
β1
]]
βπ0
.
(51)
If π = 1; πΌ = 2; and π½ = 1, (51) reduces to the fatigue life ππ
for a Neo-Hookean incompressible model
ππ = (π0 + 1)β1
[
ππ
2(π β1
0)
Γ [ ((1 + Ξπ)2β (1 + Ξπ)
β1)
Γ 2 ((1 + Ξπ) β (1 + Ξπ)β2
)
Γ(2 (1 + Ξπ) + (1 + Ξπ)β2
)β1
] ]
βπ0
.
(52)
Based on the experimental results and curve fitting, π0 and π 0
are obtained as 5.54 and 6.83MPa, respectively.
4. Simulation and Experimental Test
Testing is an important part of designing components foracceptable fatigue life. The analysis in the design phase isused to obtain dimensions that are most likely to providedesired results and to minimize costly and time-consumingiterations in prototyping and testing. Types of testing rangeform standard fatigue specimen tests to testing the actualdevice under operating conditions.
The testing of the actual device usually requires morework than for a standard test specimen. Twomajor challengesaccompany the testing of compliantmechanisms. First, a newtest must be designed for each new type of device. Secondly,because of the large number of cycles required for fatiguetesting, the test device may also fail due to fatigue. Standardfatigue specimen test is adopted here.
4.1. Sample Preparation. Low density polypropylene wasselected as the test material. The geometry and test length ofthe specimens are as shown in Figure 3. It has a gauge lengthof 22mm.
4.2. Finite Element Analysis (FEA). In order to better under-stand the stress distributions, FEA was performed. An FEAmodel of an undamaged LDP specimen was employed. Themodel was built in solid edge ST 4. For the static FEA,ANSYS 13.0 was used as pre/post processor and solver. Thespecimen model was subjected to static and cyclic loading.Then, ANSYS nCode DesignLife was used for the fatigue lifeanalysis.
4.3. Fatigue and Uniaxial Tensile Tests. The BOSE Electro-Force (ELF) 3200 testing machine (Figure 4), in conjuctionwith the WinTest control software, was used to conduct the
10
2252
101.6
8
R5
Figure 3: Geometry of LDP test specimen.
LDP specimen
Figure 4: Experimental setup.
mechanical experiments in uniaxial cyclic loading and uniax-ial tension.The test instrument incorporate proprietary Boselinear motion technologies and WinTest controls to providea revolutionary approach to mechanical fatigue and dynamiccharacterization. The ELF has a maximum load of 225N anda maximum frequency of 400Hz. A set of low mass grips,model GRP-TCDMA450N from BOSE ElectroForce (EdenPrairie, MN, USA), were used. The load cell had a maximumload rating of 2.5N (250 g) and resolution of Β±10mg. TheELF 3200 measures displacements via a Capacitec 100 lmdisplacement transducer (model HPC-40/4101) used as afeedback for the control loop.
For the quasistatic tests, the specimens were placed undertension at a controlled rate of displacement (0.02mm/s).The clamping length was about 40mm. The mechanicalproperties, that is, Youngβs modulus, shear modulus, ultimatestrain, and stresswere assessed for the specimens.The averageof the results was taken as the resultant value. To determinelocal data (stress and strain) from global data (force anddisplacement), the specimenβs dimensionswere obtainedwiththe use of themicrometer screw gauge and the vernier caliper,and its modulus is calculated from the linear part of theresulting stress-strain curve shown in Figure 5. The material
ISRN Polymer Science 9
0 0.05 0.1 0.15 0.2 0.250
5
10
15
Strain (mm/mm)
Stre
ss (M
Pa)
Figure 5: Stress-strain curve for LDP.
parameter ππ is obtained by fitting the experimental stress-strain curve into (22). This yields ππ = 43.04MPa.
The fatigue experiments were conducted between aminimum and maximum load in tension for a prescribednumber of cycles using a set frequency. All fatigue tests wereconducted at 10Hz. The residual strain was determined byconsidering the displacement reading on the oscilloscope atthe initiation of the test and at the conclusion of the dwellphase. The residual strain was measured by subtracting theoscilloscope displacement value at the initiation of the testfrom the displacement at the conclusion of the constant-stressamplitude fatigue phase, once the specimen was unloaded tozero stress and allowed to dwell for a short time.
Approximately 40 fibers were tested in this investigation.Approximately 20% of the fibers broke at the grip
interface at the conclusion of the uniaxial tensile loadingphase, which indicated premature failure due to a stressconcentration near the grip interface. For this reason, theseexperimental results were omitted from the results in thisstudy. All fibers were subjected to the same frequency duringcyclic loading (10Hz), elongation rate during uniaxial tension(0.02mm/s).
Figure 6 is the oscilloscope output of load and displace-ment versus time response of the LDP samples, while Figures7 and 8 display their 1 s interval of the strain versus time andstress versus time response under sinusoidal cyclic loadingconditions after equilibrium stress and strain values werereached. These samples were subjected to uniaxial fatigueloading conditions. The graphs show the behaviour of thesamples under different displacement inputs.The strain rangeincreases within increase in the input displacement. Thefatigue life verses strain amplitude in a logarithmic coor-dinate is shown in Figure 9. The scattered points representexperimental results, the black dotted line is the result fromfatigue simulation software while the red dotted line is fromthe derived formula.
4.4. Hysteresis Loop. In fatigue, the area of a stress-strainhysteresis loop is a measure of mechanical energy lost due
Figure 6: Oscilloscope output of load and displacement versus timeresponse of LDP sample undergoing uniaxial sinusoidal loading.
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
Time (s)
Stra
in (m
m/m
m)
β0.15
β0.1
β0.05
0.5 disp.1.5 disp.
2.5 disp.3.5 disp.
Figure 7: A second strain cycles.
to viscoelastic damping during each cycle of extension andcompression. For a viscoelastic material subject to the cyclicloading, the hysteresis of the material can be defined byplotting the input stress π(π‘) versus the responding strainπ(π‘) for one cycle of motion. Polymeric material hysteresisloops are difficult to analyse but can reveal interesting insightinto the behaviour of the material during fatigue testing[31]. Figure 10 shows hysteresis loops of the low densitypolypropylene for 10 complete cycles. From the figure, thehysteresis curves for LDP are generally asymmetrical. Thegraph shows an early increase in maximum stress withincrease in the input displacement load before it got tothe maximum value and started decreasing. The loops hadimmediate stability for all range of displacement load input.The area captured within the hysteresis loop is equal tothe dissipation energy per cycle of harmonic motion by thematerial. Within the tensile portion of the loop, considerableplastic strain and crack propagation energy is lost as theintersection through the zero stress line is at a positive strainvalue. During compression, the intersection of the zero stressline is very close to zero, implying little plastic strain. Thisbehaviour of thematerial in compression reduces the amountof energy lost per cycle.
10 ISRN Polymer Science
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Time (s)
Stre
ss (M
Pa)
β2
0.5 disp.1.5 disp.
2.5 disp.3.5 disp.
Figure 8: A second stress cycles.
1000
10
0.05 0.1 0.15 0.2 0.25
Model fittedANSYS nCode DesignLifeExperimental
Strain amplitude (mm/mm)
1011
109
107
105
Fatig
ue li
feπ
π(c
ycle
s)
Figure 9: Fatigue life ππ versus strain amplitude.
12
10
8
6
4
2
0
β2
β0.15 β0.1 β0.05 0 0.05 0.1
3.5 disp.2.5 disp.
1.5 disp.0.5 disp.
Stre
ss ra
nge,Ξπ
(MPa
)
Strain range, Ξπ (mm/mm)
Figure 10: Hysteresis loop for the LDP material.
0 0.05 0.1 0.15 0.2 0.25
ModelExpnCode
Strain amplitude, Ξπ (mm/mm)
Fatig
ue li
feπ
π(c
ycle
s)
1πΈ+001πΈ+011πΈ+021πΈ+031πΈ+041πΈ+051πΈ+061πΈ+071πΈ+081πΈ+091πΈ+101πΈ+111πΈ+12
1πΈβ01
Figure 11: Standard error bars.
4.5. Statistical Inference Using Error Bars. Means and stan-dard error (SE) bars are shown in Figure 11 for test experi-ments where the fatigue life in ten independent fatigue testexperiments as well as the corresponding model and ANSYSnCode results of the test specimens were determined overstrain amplitude (Figure 9). Error bars were used to assessdifferences between groups at the same strain amplitudepoints. Overlap rule was used to estimate the group points.The mean and their error regions overlapped, showing thatthe groups are not statistically and significantly different fromone another.
5. Conclusion
A theoretical formula based on nominal strain amplitude forthe fatigue life prediction of compliant polymeric materialhas been presented. The fatigue life prediction formulawas developed within the context of continuum damagemechanics. Hyperelastic model is used to capture the largedeformation behaviour of polymeric compliant mechanism.The strain energy function is formulated in terms of the straininvariants.
ANSYS nCode DesignLife, that is, mainly based on thereversed stress cycle, is also used to perform the fatiguesimulation. Fatigue and tensile tests were conducted underdisplacement controlled loading condition. Nominal stress-strain curve was obtained from which the material parame-ters were determined by curve fitting.
Immediate stability in the hysteresis loops and littleplastic damage that occurs within the compressive region ofthe fatigue cycles make LDP suitable for CMs.
The statistical analysis of the theoretical prediction for-mula with the experimental data and simulation result showsa strong agreement. Therefore, the CDM-based model can
ISRN Polymer Science 11
be applied to study fatigue life for compliant systems withvarying stress and frequency.
Conflict of Interests
The authors declare that they have no conflict of interests.
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